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Vector parallel to the line of intersection of two planes, each spanned by two given vectors: angle with another vector

The same question, asked 2 times: 26 Jul 2022*, 11 Apr 2023. Every wording below, oldest first.

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  1. Question 1

    26 Jul 2022

    165
    A vector a→\overrightarrow{a} is parallel to the line of intersection of the plane determined by the vectors i^,i^+j^\widehat{i},\widehat{i}+\widehat{j} and the plane determined by the vectors i^−j^,i^+k^\widehat{i}-\widehat{j},\widehat{i}+\widehat{k}. The obtuse angle between a→\overrightarrow{a} and the vector b→=i^−2j^+2k^\overrightarrow{b}=\widehat{i}- 2\widehat{j}+ 2\widehat{k} is

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  2. Question 2

    11 Apr 2023

    78
    Let a→\overrightarrow{a} be a non-zero vector parallel to the line of intersection of the two planes determined by the vectors i^+j^,i^+k^\widehat{i}+\widehat{j},\widehat{i}+\widehat{k} and i^−j^,j^−k^\widehat{i}-\widehat{j},\widehat{j}-\widehat{k}. If θ\theta is the angle between the vector a→\overrightarrow{a} and the vector b→=2i^−2j^+k^\overrightarrow{b}= 2\widehat{i}- 2\widehat{j}+\widehat{k} and a→⋅b→=6\overrightarrow{a}\cdot\overrightarrow{b}= 6 then the ordered pair (θ,∣a→×b→∣)(\theta,|\overrightarrow{a}\times\overrightarrow{b}|) is equal to

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